arXiv subjects

Mihai-Alin Badiu

Publications and source records attributed to Mihai-Alin Badiu.

At least 19 recordsLinked to original sources

arXiv2026

On the Entropy of a Random Geometric Graph

In this paper, we study the entropy of a hard random geometric graph (RGG), a commonly used model for spatial networks, where the connectivity is governed by the distances between the nodes. Formally, given a connection range $r$, a hard RGG $G_m$ on $m$ vertices is formed by drawing $m$ random points from a spatial domain, and then connecting any two points with an edge when they are within a distance $r$ from each other. The two domains we consider are the $d$-dimensional unit cube $[0,1]^d$ and the $d$-dimensional unit torus $\mathbb{T}^d$. We derive upper bounds on the entropy $H(G_m)$ for both these domains and for all possible values of $r$. In a few cases, we obtain an exact asymptotic characterization of the entropy by proving a tight lower bound. Our main results are that $H(G_m) \sim dm \log_2m$ for $0 < r \leq 1/4$ in the case of $\mathbb{T}^d$ and that the entropy of a one-dimensional RGG on $[0,1]$ behaves like $m\log m$ for all $0<r<1$. As a consequence, we can infer that the asymptotic structural entropy of an RGG on $\mathbb{T}^d$, which is the entropy of an unlabelled RGG, is $Ω((d-1)m \log_2m)$ for $0 < r \leq 1/4$. For the rest of the cases, we conjecture that the entropy behaves asymptotically as the leading order terms of our derived upper bounds.

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arXiv2025

Rate-Distortion-Perception Function of Bernoulli Vector Sources

In this paper, we consider the rate-distortion-perception (RDP) trade-off for the lossy compression of a Bernoulli vector source, which is a finite collection of independent binary random variables. The RDP function quantifies in a way the efficient compression of a source when we impose a distortion constraint that limits the dissimilarity between the source and the reconstruction and a perception constraint that restricts the distributional discrepancy of the source and the reconstruction. In this work, we obtain an exact characterization of the RDP function of a Bernoulli vector source with the Hamming distortion function and a single-letter perception function that measures the closeness of the distributions of the components of the source. The solution can be described by partitioning the set of distortion and perception levels $(D,P)$ into three regions, where in each region the optimal distortion and perception levels we allot to the components have a similar nature. Finally, we introduce the RDP function for graph sources and apply our result to the Erdős-Rényi graph model.

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arXiv2025

Graph Compression with Side Information at the Decoder

In this paper, we study the problem of graph compression with side information at the decoder. The focus is on the situation when an unlabelled graph (which is also referred to as a structure) is to be compressed or is available as side information. For correlated Erdős-Rényi weighted random graphs, we give a precise characterization of the smallest rate at which a labelled graph or its structure can be compressed with aid of a correlated labelled graph or its structure at the decoder. We approach this problem by using the entropy-spectrum framework and establish some convergence results for conditional distributions involving structures, which play a key role in the construction of an optimal encoding and decoding scheme. Our proof essentially uses the fact that, in the considered correlated Erdős-Rényi model, the structure retains most of the information about the labelled graph. Furthermore, we consider the case of unweighted graphs and present how the optimal decoding can be done using the notion of graph alignment.

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arXiv2023

Rate-Distortion Function of the Stochastic Block Model

The stochastic block model (SBM) is extensively used to model networks in which users belong to certain communities. In recent years, the study of information-theoretic compression of such networks has gained attention, with works primarily focusing on lossless compression. In this work, we address the lossy compression of SBM graphs by characterizing the rate-distortion function under a Hamming distortion constraint. Specifically, we derive the conditional rate-distortion function of the SBM with community membership as side information. We approach this problem as the classical Wyner-Ziv lossy problem by minimising mutual information of the graph and its reconstruction conditioned on community labels. Lastly, we also derive the rate-distortion function of the Erdős-Rényi (ER) random graph model.

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arXiv2023

On Random Tree Structures, Their Entropy, and Compression

Measuring the complexity of tree structures can be beneficial in areas that use tree data structures for storage, communication, and processing purposes. This complexity can then be used to compress tree data structures to their information-theoretic limit. Additionally, the lack of models for random generation of trees is very much felt in mathematical modeling of trees and graphs. In this paper, a number of existing tree generation models such as simply generated trees are discussed, and their information content is analysed by means of information theory and Shannon's entropy. Subsequently, a new model for generating trees based on practical appearances of trees is introduced, and an upper bound for its entropy is calculated. This model is based on selecting a random tree from possible spanning trees of graphs, which is what happens often in practice. Moving on to tree compression, we find approaches to universal tree compression of the discussed models. These approaches first transform a tree into a sequence of symbols, and then apply a dictionary-based compression method. Conditions for the universality of these method are then studied and analysed.

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arXiv2023

Optimal Mobility and Communication Strategy to Maximize the Value of Information in IoT Networks

The Internet of Things (IoT) is an emerging next-generation technology in the fourth industrial revolution. In industrial IoT networks, sensing devices are largely deployed to monitor various types of physical processes. They are required to transmit the collected data in a timely manner to support real-time monitoring, control and automation. The timeliness of information is very important in such systems. Recently, an information-theoretic metric named the "value of information" (VoI) has been proposed to measure the usefulness of information. In this work, we consider an industrial IoT network with a set of heterogeneous sensing devices and an intelligent mobile entity. The concept of the value of information is applied to study a joint path planning and user scheduling optimisation problem. We aim to maximise the network-level VoI under mobility and communication constraints. We formulate this problem as a Markov decision process (MDP), and an efficient algorithm based on reinforcement learning is proposed to solve this problem. Through numerical results, we show that the proposed method is able to capture the usefulness of data from both time and space dimensions. By exploiting the correlation property of the data source, the proposed method is suitable for applications in resource-limited networks.

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arXiv2022

LEAST: a Low-Energy Adaptive Scalable Tree-based routing protocol for Wireless Sensor Networks

Routing is one of the critical and ongoing challenges in Wireless Sensor Networks. The main challenge has always been to have a routing protocol that reduces the communication overhead, hence saving the energy of the sensors in the network. Hierarchical routing protocols are known to be the most energy-efficient routing protocols for Wireless Sensor Networks. In this paper, a more generalized hierarchical routing protocol is introduced for Wireless Sensor Network, which is based on tree data structures. The clustering in the proposed protocol has the format of a general tree, which is constructed in an adaptive manner based on the distance of the sensors. Results show that the proposed tree-based protocol introduces a significant benefit in energy consumption and lifetime of the network over the existing hierarchical approaches.

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arXiv2022

Performance Analysis of IOS-Assisted NOMA System with Channel Correlation and Phase Errors

In this paper, we investigate the performance of an intelligent omni-surface (IOS) assisted downlink non-orthogonal multiple access (NOMA) network with phase quantization errors and channel estimation errors, where the channels related to the IOS are spatially correlated. First, upper bounds on the average achievable rates of the two users are derived. Then, channel hardening is shown to occur in the proposed system, based on which we derive approximations of the average achievable rates of the two users. The analytical results illustrate that the proposed upper bound and approximation on the average achievable rates are asymptotically equivalent in the number of elements. Furthermore, it is proved that the asymptotic equivalence also holds for the average achievable rates with correlated and uncorrelated channels. Additionally, we extend the analysis by evaluating the average achievable rates for IOS assisted orthogonal multiple access (OMA) and IOS assisted multi-user NOMA scenarios. Simulation results corroborate the theoretical analysis and demonstrate that: i) low-precision elements with only two-bit phase adjustment can achieve the performance close to the ideal continuous phase shifting scheme; ii) The average achievable rates with correlated channels and uncorrelated channels are asymptotically equivalent in the number of elements; iii) IOS-assisted NOMA does not always perform better than OMA due to the reconfigurability of IOS in different time slots.

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arXiv2022

Performance Analysis of RIS-Assisted Large-Scale Wireless Networks Using Stochastic Geometry

In this paper, we investigate the performance of a reconfigurable intelligent surface (RIS) assisted large-scale network by characterizing the coverage probability and the average achievable rate using stochastic geometry. Considering the spatial correlation between transmitters (TXs) and RISs, their locations are jointly modelled by a Gauss-Poisson process (GPP). Two association strategies, i.e., nearest association and fixed association, are both discussed. For the RIS-aided transmission, the signal power distribution with a direct link is approximated by a gamma random variable using a moment matching method, and the Laplace transform of the aggregate interference power is derived in closed form. Based on these expressions, we analyze the channel hardening effect in the RIS-assisted transmission, the coverage probability, and the average achievable rate of the typical user. We derive the coverage probability expressions for the fixed association strategy and the nearest association strategy in an interference-limited scenario in closed form. Numerical results are provided to validate the analysis and illustrate the effectiveness of RIS-assisted transmission with passive beamforming in improving the system performance. Furthermore, it is also unveiled that the system performance is independent of the density of TXs with the nearest association strategy in the interference-limited scenario.

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arXiv2022

TreeExplorer: a coding algorithm for rooted trees with application to wireless and ad hoc routing

Routing tables in ad hoc and wireless routing protocols can be represented using rooted trees. The constant need for communication and storage of these trees in routing protocols demands an efficient rooted tree coding algorithm. This efficiency is defined in terms of the average code length, and the optimality of the algorithm is measured by comparing the average code length with the entropy of the source. In this work, TreeExplorer is introduced as an easy-to-implement and nearly optimal algorithm for coding rooted tree structures. This method utilizes the number of leaves of the tree as an indicator for choosing the best method of coding. We show how TreeExplorer can improve existing routing protocols for ad hoc and wireless systems, which normally entails a significant communication overhead.

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arXiv2022

Structural Complexity of One-Dimensional Random Geometric Graphs

We study the richness of the ensemble of graphical structures (i.e., unlabeled graphs) of the one-dimensional random geometric graph model defined by $n$ nodes randomly scattered in $[0,1]$ that connect if they are within the connection range $r\in[0,1]$. We provide bounds on the number of possible structures which give universal upper bounds on the structural entropy that hold for any $n$, $r$ and distribution of the node locations. For fixed $r$, the number of structures is $Θ(a^{2n})$ with $a=a(r)=2 \cos{\left(\fracπ{\lceil 1/r \rceil+2}\right)}$, and therefore the structural entropy is upper bounded by $2n\log_2 a(r) + O(1)$. For large $n$, we derive bounds on the structural entropy normalized by $n$, and evaluate them for independent and uniformly distributed node locations. When the connection range $r_n$ is $O(1/n)$, the obtained upper bound is given in terms of a function that increases with $n r_n$ and asymptotically attains $2$ bits per node. If the connection range is bounded away from zero and one, the upper and lower bounds decrease linearly with $r$, as $2(1-r)$ and $(1-r)\log_2 e$, respectively. When $r_n$ is vanishing but dominates $1/n$ (e.g., $r_n \propto \ln n / n$), the normalized entropy is between $\log_2 e \approx 1.44$ and $2$ bits per node. We also give a simple encoding scheme for random structures that requires $2$ bits per node. The upper bounds in this paper easily extend to the entropy of the labeled random graph model, since this is given by the structural entropy plus a term that accounts for all the permutations of node labels that are possible for a given structure, which is no larger than $\log_2(n!) = n \log_2 n - n + O(\log_2 n)$.

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arXiv2022

A Framework for Characterising the Value of Information in Hidden Markov Models

In this paper, a general framework is formalised to characterise the value of information (VoI) in hidden Markov models. Specifically, the VoI is defined as the mutual information between the current, unobserved status at the source and a sequence of observed measurements at the receiver, which can be interpreted as the reduction in the uncertainty of the current status given that we have noisy past observations of a hidden Markov process. We explore the VoI in the context of the noisy Ornstein-Uhlenbeck process and derive its closed-form expressions. Moreover, we investigate the effect of different sampling policies on VoI, deriving simplified expressions in different noise regimes and analysing statistical properties of the VoI in the worst case. We also study the optimal sampling policy to maximise the average information value under the sampling rate constraint. In simulations, the validity of theoretical results is verified, and the performance of VoI in Markov and hidden Markov models is also analysed. Numerical results further illustrate that the proposed VoI framework can support timely transmission in status update systems, and it can also capture the correlation properties of the underlying random process and the noise in the transmission environment.

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arXiv2021

BernoulliZip: a Compression Algorithm for Bernoulli Processes and Erdos-Renyi Graphs

A novel compression scheme for compressing the outcome of $n$ independent Bernoulli trials is introduced and analysed. The resulting algorithm, BernoulliZip, is a fast and near-optimal method to produce prefix codes for a Bernoulli process. BernoulliZip's main principle is to first represent the number of 1s in the sequence and then specify the sequence. The application of BernoulliZip on compressing Erdos-Renyi graphs is explored.

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arXiv2020

Study of Intelligent Reflective Surface Assisted Communications with One-bit Phase Adjustments

We analyse the performance of a communication link assisted by an intelligent reflective surface (IRS) positioned in the far field of both the source and the destination. A direct link between the transmitting and receiving devices is assumed to exist. Perfect and imperfect phase adjustments at the IRS are considered. For the perfect phase configuration, we derive an approximate expression for the outage probability in closed form. For the imperfect phase configuration, we assume that each element of the IRS has a one-bit phase shifter (0°, 180°) and an expression for the outage probability is obtained in the form of an integral. Our formulation admits an exact asymptotic (high SNR) analysis, from which we obtain the diversity orders for systems with and without phase errors. We show these are N + 1 and (N + 3)/2, respectively. Numerical results confirm the theoretical analysis and verify that the reported results are more accurate than methods based on the central limit theorem (CLT).

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arXiv2020

A Value of Information Framework for Latent Variable Models

In this paper, a general value of information (VoI) framework is formalised for latent variable models. In particular, the mutual information between the current status at the source node and the observed noisy measurements at the destination node is used to evaluate the information value, which gives the theoretical interpretation of the reduction in uncertainty in the current status given that we have measurements of the latent process. Moreover, the VoI expression for a hidden Markov model is obtained in this setting. Numerical results are provided to show the relationship between the VoI and the traditional age of information (AoI) metric, and the VoI of Markov and hidden Markov models are analysed for the particular case when the latent process is an Ornstein-Uhlenbeck process. While the contributions of this work are theoretical, the proposed VoI framework is general and useful in designing wireless systems that support timely, but noisy, status updates in the physical world.

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arXiv2020

Chernoff Bounds and Saddlepoint Approximations for the Outage Probability in Intelligent Reflecting Surface Assisted Communication Systems

We analyze the outage probability of an intelligent reflecting surface (IRS)-assisted communication network. A tight upper bound on the outage probability is formulated based on the Chernoff inequality. Furthermore, through an exact asymptotic (a large number of reflecting elements) analysis based on a saddlepoint approximation, we derive closed-form expressions of the outage probability for systems with and without a direct link and obtain the corresponding diversity orders. Simulation results corroborate our theoretical analysis and show the inaccuracies inherent in using the central limit theorem (CLT) to analyze system performance. Our analysis is accurate even for a small number of IRS elements in the high signal-to-noise ratio (SNR) regime.

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arXiv2019

Statistical Properties of Transmissions Subject to Rayleigh Fading and Ornstein-Uhlenbeck Mobility

In this paper, we derive closed-form expressions for significant statistical properties of the link signal-to-noise ratio (SNR) and the separation distance in mobile ad hoc networks subject to Ornstein-Uhlenbeck (OU) mobility and Rayleigh fading. In these systems, the SNR is a critical parameter as it directly influences link performance. In the absence of signal fading, the distribution of the link SNR depends exclusively on the squared distance between nodes, which is governed by the mobility model. In our analysis, nodes move randomly according to an Ornstein-Uhlenbeck process, using one tuning parameter to control the temporal dependency in the mobility pattern. We derive a complete statistical description of the squared distance and show that it forms a stationary Markov process. Then, we compute closed-form expressions for the probability density function (pdf), the cumulative distribution function (cdf), the bivariate pdf, and the bivariate cdf of the link SNR. Next, we introduce small-scale fading, modelled by a Rayleigh random variable, and evaluate the pdf of the link SNR for rational path loss exponents. The validity of our theoretical analysis is verified by extensive simulation studies. The results presented in this work can be used to quantify link uncertainty and evaluate stability in mobile ad hoc wireless systems.

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arXiv2019

Communication Through a Large Reflecting Surface With Phase Errors

Assume the communication between a source and a destination is supported by a large reflecting surface (LRS), which consists of an array of reflector elements with adjustable reflection phases. By knowing the phase shifts induced by the composite propagation channels through the LRS, the phases of the reflectors can be configured such that the signals combine coherently at the destination, which improves the communication performance. However, perfect phase estimation or high-precision configuration of the reflection phases is unfeasible. In this paper, we study the transmission through an LRS with phase errors that have a generic distribution. We show that the LRS-based composite channel is equivalent to a point-to-point Nakagami fading channel. This equivalent representation allows for theoretical analysis of the performance and can help the system designer study the interplay between performance, the distribution of phase errors, and the number of reflectors. Numerical evaluation of the error probability for a limited number of reflectors confirms the theoretical prediction and shows that the performance is remarkably robust against the phase errors.

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